The speed of light is the maximum velocity that any mass can approach. In the situation you describe with the spacecraft, no matter how much time the spacecraft has to accelerate or how large the star is, it is still not enough. The problem is that even though the spaceship would initially accelerate very quickly, the acceleration would slow down near the speed of light. Einstein showed in his theory of special relativity that the mass of an object increases very rapidly as its velocity approaches the speed of light. In fact, the relativistic mass becomes infinite at the speed of light. This also means that the energy of the spacecraft approaches infinity from Einstein's equation (E=mc²). Since the star would need to have infinite mass (and it won't) to accelerate an object of infinite mass, it will never reach the speed of light.